Traffic measurements from communication networks have shown that many quantities characterizing network performance have long-tail probability distributions, i.e., with tails that decay more slowly than exponentially. File lengths, call holding times, scene lengths in MPEG video streams, and intervals between connection requests in Internet traffic all have been found to have long-tail distributions, being well described by distributions such as the Pareto and Weibull. It is known that long-tail distributions can have a dramatic effect upon performance, e.g., long-tail service-time distributions cause long-tail waiting-time distributions in queues, but it is often difficult to describe this effect in detail, because performance models with component long-tail distributions tend to be difficult to analyze. We address this problem by developing an algorithm for approximating a long-tail distribution by a hyperexponential distribution (a finite mixture of exponentials). We first prove tha...

The main objective of this sequel is to solve the out-of-sequence problem that occurs in the load balanced Birkhoff-von Neumann switch with one-stage buffering. We do this by adding a load-balancing buffer in front of the first stage and a resequencing-and-output buffer after the second stage. Moreover, packets are distributed at the first stage according to their flows, instead of their arrival times in Part I. In this paper, we consider multicasting ows with two types of scheduling policies: the First Come First Served (FCFS) policy and the Earliest Deadline First (EDF) policy. The FCFS policy requires a jitter control mechanism in front of the second stage to ensure proper ordering of the traffic entering the second stage. For the EDF scheme, there is no need for jitter control. It uses the departure times of the corresponding FCFS output-buffered switch as deadlines and schedules packets according to their deadlines. For both policies, we show that the end-to-end delay through our multistage switch is bounded above by the sum of the delay from the corresponding FCFS output-buffered switch and a constant that only depends on the size of the switch and the number of multicasting flows supported by the switch.

Consider an aggregate arrival process A N obtained by multiplexing N On-Off processes with exponential Off periods of rate and subexponential On periods ø on . As N goes to infinity, with N ! , A N approaches an M=G=1 type process. Both for finite and infinite N , we obtain the asymptotic characterization of the arrival process activity period. Using these results we investigate a fluid queue with the limiting M/G/1 arrival process A 1 t and capacity c. When On periods are regularly varying (with non-integer exponent), we derive a precise asymptotic behavior of the queue length random variable Q P t observed at the beginning of the arrival process activity periods P[Q P t ? x] ¸ r + ae \Gamma c c \Gamma ae Z 1 x=(r+ae\Gammac) P[ø on ? u]du x !1; where ae = EA 1 t ! c; r (c r) is the rate at which the fluid is arriving during an On period. The asymptotic (time average) queue distribution lower bound is obtained under more general assumptions on On periods than reg...

The Effect of Multiple Time Scales and Subexponentiality on the Behavior of a Broadband Network Multiplexer Predrag R. Jelenkovi'c The main theme of this dissertation is the evaluation of the capacity of broadband multimedia network multiplexers. This problem calls for the modeling of network traffic streams and the analysis of a network multiplexer that is loaded with the corresponding models. For modeling we focus on MPEG video traffic streams that are expected to be predominant in the traffic mixture of future multimedia networks. We experimentally demonstrate that real-time MPEG video traffic exhibits multiple time scale characteristics, as well as subexponential first and second order statistics. Then we construct a model of MPEG video that captures both of these characteristics and accurately predicts queueing behavior for a broad range of buffer and capacity sizes. Depending on whether a network multiplexer (loaded with MPEG) is strictly or weakly stable the dominant effect o...

We study the tail asymptotics of the r.v. X(T ) where fX(t)g is a stochastic process with a linear drift and satisfying some regularity conditions like a central limit theorem and a large deviations principle, and T is an independent r.v. with a subexponential distribution. We find that the tail of X(T ) is sensitive to whether or not T has a heavier or lighter tail than a Weibull distribution with tail e \Gamma p x . This leads to two distinct cases, heavy-tailed and moderately heavy-tailed, but also some results for the classical light-tailed case are given. The results are applied via distributional Little's law to establish tail asymptotics for steady--state queue length in GI/GI/1 queues with subexponential service times. Further applications are given for queues with vacations, and M/G/1 busy periods.

This is a survey paper on fluid queues, with a strong emphasis on recent attempts to represent phenomena like long-range dependence. The central model of the paper is a fluid queueing system fed by N independent sources that alternate between silence and activity periods. The distribution of the activity periods of at least one source is assumed to be long-tailed, which may give rise to long-range dependence. We consider the effect of this tail behaviour on the steady-state distributions of the buffer content at embedded points in time and at arbitrary time, and on the busy period distribution. Both exact results and bounds are discussed.

We survey the properties and uses of the class of subexponential probability distributions, paying particular attention to their use in modelling heavy-tailed data such as occurs in insurance and queueing applications. We give a detailed summary of the core theory and discuss subexponentiality in various contexts including extremes, random walks and L'evy processes with negative drift, and sums of random variables, the latter extended to cover random sums, weighted sums and moving averages. 1. Definition and first properties Subexponential distributions are a special class of heavy--tailed distributions. The name arises from one of their properties, that their tails decrease more slowly than any exponential tail; see (1.4). This implies that large values can occur in a sample with non--negligible probability, and makes the subexponential distributions candidates for modelling situations where some extremely large values occur in a sample compared to the mean size of the data. Such a p...

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Consider an aggregate arrival process A N obtained by multiplexing N On-Off sources with exponential Off periods of rate and subexponential On periods ø on . For this process its activity period I N satisfies P[I N ? t] ¸ (1 + Eø on ) N \Gamma1 P[ø on ? t] as t !1; for all sufficiently small . When N goes to infinity, with N ! , A N approaches an M=G=1 type process, for which the activity period I 1 , or equivalently a busy period of an M=G=1 queue with subexponential service requirement ø on , satisfies P[I 1 ? t] ¸ e Eø on P[ø on ? t] as t !1. For a simple subexponential On-Off fluid flow queue we establish a precise asymptotic relation between the Palm queue distribution and the time average queue distribution. Further, a queueing system in which one On-Off source, whose On period belongs to a subclass of subexponential distributions, is multiplexed with independent exponential sources with aggregate expected rate Ee t , is shown to be asymptotically ...

We develop a practical, multiple time scale model for MPEG video traffic whose accuracy and relatively low computational complexity make it well suited for real-time traffic generation experiments on broadband networks. The major feature of our approach is the decomposition of the frame size sequence into simple slow and fast time scale components. This accurately captures aspects of queueing behavior that are difficult to model otherwise. The model also exploits the existence of deterministic patterns that are due to the MPEG coding scheme. We also present a novel modeling approach based on spatial renewal processes (SRP). This model gives exact matches to any desired marginal distribution and any convex non-increasing autocorrelation function. In particular, it can match subexponentially decaying autocorrelations (i.e., can capture long range dependence), something no other model of comparable complexity can do. A SRP is suited for on-line model construction, since it involves no sea...